Log-Sum Penalized Differential Graph Estimation from Time-Dependent Data

Estimation of differences in conditional independence graphs (CIGs) of two time series Gaussian graphical models (TSGGMs) is investigated where the two TSGGMs are known to have similar structure. The TSGGM structure is encoded in the inverse power spectral density (IPSD) of the time series. In several existing works, one is interested in estimating the difference in two precision matrices to characterize underlying changes in conditional dependencies of two sets of data consisting of independent and identically distributed (i.i.d.) observations. In this paper we consider estimation of the difference in two IPSDs to characterize the underlying changes in conditional dependencies of two sets of time-dependent data. We extend an existing group lasso-penalized D-trace loss function approach in the frequency domain for differential time-series graph learning to non-convex group log-sum penalized D-trace loss function approach to improve performance. An alternating direction method of multipliers (ADMM) algorithm is presented to optimize the objective function. A synthetic data example is presented in support of the proposed approach where our proposed log-sum penalized loss significantly outperforms existing approaches with $F_{1}$ score as the performance metric.